Maths - Finite Groups

In
a finite group there are discreet or finite steps between elements of
the group. We cannot move continuously between them as we can in
infinite groups.

Permutation
group

The permutation group seems to me to have a sort of indirection about it. We start with a set, we define some permutations of the set, we then treat these permutations as another set and this set of permutations are the elements of
the group.

So a permutation group is defined as group G whose elements are permutations of a given set M and whose
group operation is the composition of permutations in G.

A
permutation (in this context - it is defined differently from the way
the term is used in probability which involves the linear order) is a
1:1 mapping from a group to itself. that is each element of the set
maps to one other (or possibly the same) element of that set. In
technical terms: this is a bijection from a finite set to itself.

Cayley's Theorem

Every group can be interpreted as a permutation group.

Cycle
Decomposition

We
could define each permutation by how it maps each element such as:

a
-> b
b
-> c
c
-> d
d
-> a

and
so on. However we could save space by combining them on a single line
as follows:

a
-> b -> c -> d -> a

This
is quite a simple case where all the elements cycle round. The
opposite extreme to this would be the null element where each item
stays the same:

a
-> a
b
-> b
c
-> c
d
-> d

In
the more general case there may be a number of cycles. such as :

a
-> b -> a
c
-> d -> c

There
is a notation that allows us to write this even more compactly. The
above case could be written:

(ab)(cd)

That
is, each cycle is shown in a separate set of brackets.

Group
Elements

Each
of the permutations is an element in the group. It is almost like
indirection in that we can define each permutation then we can define
how they interact under composition.

Example
1

e
= (1)(2)(3)(4)
a
= (1 2)(3)(4)
b
= (1)(2)(3 4)
a
o b = (1 2)(3 4)

Example
2 - Triangle

In this example lets take the rotation of an equilateral triangle to itself. This has 3 possible rotations:

identity (0°)

anticlockwise (120°)

clockwise (240°)

permutation

cycle notation

(1)(2)(3)

(1 2 3)

(3 2 1)

These letters then become the elements of a set of permutations. We can then work out the result of the composition of these permutations:

o

=

To be consistent with other literature on this, we need to reverse the order of the functions, that is the right hand function is applied first then the left hand function. That is:

(d o c)(x) = d(c(x))

where:

d(x) = apply the 'd' permutation to x

c(x) = apply the 'c' permutation to x

o = composition operation

(d o c)(x) = apply the composition of c then d to x

We can draw up a complete table of the composition operation which defines this example completely:

row o col

i

c

d

i

c

d

We can also work out the inverse of each permutation:

x

x-1

We can extend this group to include reflections, see this page for more.

Example
3 - Cube

Lets
take as an example the rotation of a cube in ways that does not
change its shape (i.e. multiples of 90° rotations about x, y or z)
although we will keep track of the rotations possibly by markings on
the faces of the cube. see this page.

Symmetric
group

The
group of all permutations is the symmetric group. The term
permutation group is usually used for a subgroup of the symmetric
group.

The number of possible ways to order the set is n factorial as we can see from this table:

number of elements of a set

possible ways to order set

1

1

2

2

3

6

4

24

n

!n

There is a 1:1 correspondence between the ordering of the set and mappings between the set and itself. For instance, in the table below, under reordering we show the elements of the set being rearranged and under mapping we show each element being mapped to another element:

Reordering

Mapping

But these are both ways of denoting the same set, i.e. each element of the set maps to another (or possibly same) element of the set.

We can assign a letter to each of these mappings:

permutation

cycle notation

(1)(2)(3)

(1)(2 3)

(1 2)(3)

(1 2 3)

(3 2 1)

(1 3)(2)

These letters then become the elements of a set of permutations. We can then work out the result of the composition of these permutations:

o

=

To be consistent with other literature on this, we need to reverse the order of the functions, that is the right hand function is applied first then the left hand function.

We can draw up a complete table of the composition operation which defines this example completely:

row o col

i

a

b

c

d

e

i

a

b

c

d

e

We can also work out the inverse of each permutation:

x

x-1

Conjugacy
Classes

Any
group H splits up into subsets C with the following properties.

Each
subset C is obtained in the following way: you take any group
element, say x. Then you take all of the elements of the group, call
them g's and form the group products g o x o g-1. Notice that x
itself is one of these products, because e o x o e-1= x. The subset
that consists of all these g o x o g-1's is one of the C's. For
example, x is an element of the C that you get by starting with x.
It does not matter which x in C you start with. You get the same
bunch of elements, namely C.

Any
two elements of C have the same character value under every
representation r. In symbols: if x and y are in the same C, then
x(x)=x(y).

These
subsets C are called the 'conjugacy classes of H' The C's are often
quite large. The larger they are, the more the group law of H fails
to be commutative.

Set Definition

A
set is a collection of things, which are called the elements of the
set.

Function Definition

A
function from a set A to a set B is a rule that assigns to each
element in A an element of B. If f is the name of the function and a is an element of A then we write f(a) to mean the
element of B that is assigned to a. A function f is often written as f: A →B.

1:1 Correspondence Definition

A
one to one correspondence from a set A to a set B is a rule that
associates to each element in A exactly one element in B, in such a
way that each element in B gets used exactly once and for exactly one
element in A.

Duality

Symmetry

Symmetry is an important topic for maths and physics.

Symmetry is important for many branches of mathematics including geometry (see this page) and group theory (see this page). Its importance can become apparent in unexpected places, for example, solving quintic equations.

We say that an object is symmetric, with respect to a given mathematical operation, if this operation does not change the object.

Nothers Theorem (discussed further on this page) says that, for every symmetry exhibited by a physical law,
there is a corresponding observable quantity that is conserved. Virtually every theory, including relativity and quantum physics is based on symmetry principles.

Where I can, I have put links to Amazon for books that are relevant to
the subject, click on the appropriate country flag to get more details
of the book or to buy it from them.

Fearless Symmetry - This
book approaches symmetry from the point of view of number theory. It
may not be for you if you are only interested in the geometrical
aspects of symmetry such as rotation groups but if you are interested
in subjects like modulo n numbers, Galois theory, Fermats last
theorem, to name a few topics the chances are you will find this book
interesting. It is written in a friendly style for a general audience
but I did not find it dumbed down. I found a lot of new concepts to
learn. It certainly gives a flavor of the complexity of the subject
and some areas where maths is still being discovered.